What is the third proportional to the numbers 10 and 25 (that is, 10 : 25 = 25 : x)?

Introduction / Context:
This question tests the concept of third proportional in ratio and proportion. When two numbers are given, the third proportional is defined using a proportional relationship. It is a standard topic in quantitative aptitude and appears frequently in competitive exams. The numbers 10 and 25 are given, and we are asked to find the value of x such that the proportional relationship is satisfied.

Given Data / Assumptions:
• First number = 10. • Second number = 25. • Third proportional = x such that 10 : 25 = 25 : x. • All numbers are assumed to be real positive quantities.

Concept / Approach:
If a, b and c are in continued proportion (or if c is the third proportional to a and b), then a : b = b : c. In terms of fractions, a / b = b / c. From this, we get c = b^2 / a. The question is therefore a direct application of this formula. We simply substitute a = 10 and b = 25 and compute c. This avoids any unnecessary complication and is the fastest way to solve such questions.

Step-by-Step Solution:
Step 1: Write the proportion according to the definition of third proportional: 10 : 25 = 25 : x. Step 2: Convert the proportion into an equation using fractions: 10 / 25 = 25 / x. Step 3: Cross multiply to remove the denominators: 10 * x = 25 * 25. Step 4: Compute 25 * 25 = 625. Step 5: So 10x = 625. Divide both sides by 10 to get x = 625 / 10. Step 6: Simplify x = 62.5.

Verification / Alternative Check:
We can check whether the original proportion holds with x = 62.5. Compute 10 : 25 which simplifies to 2 : 5. Now compute 25 : 62.5. Divide both numbers by 12.5 to simplify. We get 25 / 12.5 = 2 and 62.5 / 12.5 = 5, so 25 : 62.5 = 2 : 5. Both ratios match, which means the proportion is correct and x = 62.5 is confirmed as the third proportional.

Why Other Options Are Wrong:
• 125 and 150 are larger numbers that might come from squaring incorrectly or misplacing terms in the proportion, but they do not satisfy 10 / 25 = 25 / x. • 225 is another distractor that also fails the proportional test when substituted for x.

Common Pitfalls:
A common mistake is to confuse third proportional with mean proportional (geometric mean). The mean proportional m between a and b satisfies a : m = m : b, which is a different relation. Another pitfall is to invert the proportion and write 25 : 10 = 25 : x, which gives the wrong formula. Remember that the pattern for third proportional is a : b = b : c, and the formula c = b^2 / a follows directly from that.

Final Answer:
The third proportional to 10 and 25 is 62.5.